# FAIR VALUE OF LOANS AND THE SPECIAL CASE OF DEVELOPMENT INSTITUTIONS

## The Argument around the Fair Valuing of Loans

Among the many differences between a loan and a bond (we have mentioned in Section 3.1.1 how loans are over-the-counter instruments with a smaller number of variations than bonds, which can be exchange traded but come in different flavors), the one we are going to focus on is the fact that bonds are securitized instruments traded in the secondary market. The only secondary activity affecting loans, and only a small section of them, is loan syndication, which is a bespoke type of trading. This is very important when discussing the action of taking the fair value of financial instruments. Let us now remind ourselves of what we mean by fair value.

To fair value an instrument means to find a value that is fair. In order for something to be considered fair we must have this be the opinion of a collection of different parties. When it comes to financial instruments, the value that is fair is the value at which someone is willing to sell and someone else is willing to buy. In order to find a tradable value we need not only calculate the appropriate cash flows (for which the simple Equations 3.1 and 3.2 would suffice), we also need to find its present value, so that we know how much it is worth at the moment we pass it on/receive it to/from someone else. Not only this, we should also be able, since both loans and bonds carry credit risk, to assess the risk associated with the borrower, calculating in practice a risky present value. The fact that a fair value can be tradable means that the models and mechanisms used to obtain it fall within the risk-neutral framework.

There has been a general consensus as to whether derivatives should be fair valued since at least 1998 (see FAS 133 [39]). Few people would now argue against the soundness of the principle behind it. Whether loans should also be fair valued is a very different case, with sound arguments (and reasonable supporters) on both sides.

For the lender a loan is an asset. Let us consider the situation of an average bank extending a variable loan of, say, $1M to an individual or an institution. The bank would first consider the credit-worthiness of the borrower and, should it then decide to extend the loan, it would book the loan as an asset worth $1M. The risk against default would be taken into account with a *loan loss provision,* a sort of valuation adjustment on the loan value but something that is far from a calculation carried in a risk- neutral framework. The argument is, should this value be enough or should the loan be fair valued with a proper^{[1]} calculation of the present value and an inclusion of the survival probability of the borrower?

One argument claims that if the loan is to be kept on the bank's book till maturity there is no real need of a fair value. After all, a way of defining fair value is as tradable value; if there is not going to be a secondary trade, what is the need for a fair value?

The opposite argument says that the goal is to assess correctly the value of the balance sheet of an institution. Let us imagine that the lending institution is bought by another one or it needs to be liquidated. Isn't it important to have a fair assessment of its assets? The fact that the drive toward fair value is driven by accounting concerns (note that the decision to fair value derivatives was agreed to by the Financial Accounting Standards Board) might mean that in the end the fair valuing of loans argument is going to prove decisive. At the time of writing, no final decision has been made yet.

With the view that one day the fair value of loans argument is going to be generally accepted, let us observe what fair valuing entails in practice. In Equation 3.1 we have given the definition of a loan by showing its general cash flow structure, but we did not show its present fair value. In order to do so we need to write it as

(3.23)

The first term in Equation 3.23 is very similar to Equation 3.1 except for the fact that we now include the discount factor *D*i and the survival probability *S*i, that is, the probability that the borrower has not defaulted before time *T*i. The first term basically states that all loan interest payments and principal repayments are subject to the borrower being solvent.

The second term states what happens in the case of default: should the borrower default between times *T*j-1 and *T*j, the lender receives the recovery rate of the loan (whatever that might mean) applied to the outstanding principal.

It is when discussing default that we need to state that, in the case of loans, there are two common practices: one is to structure loans so that they are *accelerating* and the other so that they are *nonaccelerating.* The former means that in case of default the maturity at time T of the loan is brought forward and the recovery rate is to be applied to the entire outstanding principal. The loan shown in Equation 3.23 is an example of an accelerating one. In the case of a nonaccelerating loan it is assumed that although defaulted, the maturity of the loan remains unchanged: the fair value of nonaccelerating loans needs to take this into account and therefore apply the recover rate not to the entire principal but simply to its present value.

The fair value of a nonaccelerating loan could be written as

(3.24)

where the content of the square bracket in the second term is indeed the fair value, at time of default, of the remaining principal payments. In the case of development organizations we shall return to the difference between accelerating and nonaccelerating loans when discussing the practice of converting loans through swaps.

For both types of loans we immediately see that fair valuing a loan changes its value dramatically (which is probably behind some of the resistance on the part of financial institutions to move toward the fair value of loans): a loan of $1M will not be worth its principal amount anymore. We have said that the effect of discounting will not impact as much as the effect of taking into account the credit risk of the borrower. We have seen in Section 3.2.1 that the survival probability *S*t must be less than unity for any future time: this means that (see Section 3.2.2) the riskier the borrower, the smaller the survival probability and the smaller the fair value of the loan, or, to put it differently, the greater the difference between calculating and not calculating the fair value of the loan.

Let us, as promised earlier, focus briefly on the recovery value *R.* In the introduction to this chapter we mentioned how development institutions can see defaults in a different way from traditional financial institutions. This is because development banks often have *preferential creditor status,* meaning that, in case of default, they are the first to be repaid. Argentina, after the 2002 default, repaid in full the loans from the IMF and the World Bank, meaning that, for all intents and purposes, it did not default on those. If this seems unfair toward the other creditors, let us not forget that not only do development institutions offer terms of lending much more favorable than normal market participants, but they are also the only ones who, during a crisis, willingly and actively lend. All this has to be compensated somehow and the preferential creditor status is a way to do so.

Because of preferential creditor status, the value of *R* in Equations 3.23 and 3.24 for a development institution are necessarily higher than for the average market participants. It cannot be 1 because, although in the example of Argentina there was full repayment, we need to view *R* as an expected value and therefore the average of multiple scenarios. In practice it can be double the amount the average market player would use.

Let us stress an important point: for a development institution the value of *R* used to obtain the survival probabilities *S,* through the methodology outlined in Section 3.2.2 is the same as for the average investment bank. When it comes, however, to valuing the recovery amount of a loan in case of default (the second term of Equations 3.23 and 3.24), then it takes a higher value. This is because everyone has to agree on the idea that default has taken place or is about to (and we know that, in the bootstrapping process, the recovery rate affects the value of Si), however, once the default has taken place, an investment bank and a development bank will view the remains in a different fashion.

Fair valuing a loan is also crucial if we want to treat its prepayment and the option embedded in it, which we mentioned in Section 3.1.1. It is in this respect that we see the biggest difference between a normal financial institution and a development organization. We shall see how this works in practice in the next section.

- [1] A proper calculation of the present value would probably impact less than a correct measure of risk: in this example, which covers the case of the majority of loans, we have considered a variable rate loan. We have seen in Section 2.2.5.2 that, in general, a floating leg will price more or less at par, meaning that a riskless variable loan of $1M valued at $1M is not such a gross approximation.